Optimal Threshold Signatures in Bitcoin

It is pointed out that these probabilities, denoted as $p(\tau)$ and $q(\tau)$ respectively, are not independent as they both operate within the same underlying threshold system. Despite this interdependence, for the sake of simplicity, the paper assumes independence between these probabilities. This simplification sets the stage for further exploration, albeit with the caveat that a more realistic model would account for the correlation between these two probabilities. The conversation suggests that incorporating this correlation could lead to a more accurate depiction of the security dynamics at play.

There is a mention of prior research on threshold signature schemes, emphasizing their design and security aspects. However, it appears there is a gap in literature specifically focusing on the statistical determination of optimal thresholds for enhancing both security and usability. This identifies an area ripe for investigation, suggesting that a more nuanced analysis could yield significant benefits in terms of securing transactions while also ensuring accessibility for users.

The upgrade to Bitcoin's protocol through Taproot is highlighted as a significant advancement, particularly for degraded multisignature transactions. Prior to Taproot, implementing such transactions was possible but less efficient in terms of fees and privacy. Taproot’s introduction marks a pivotal enhancement, improving both cost-efficiency and privacy for users engaging in complex transactions. This evolution underscores Bitcoin's ongoing development towards facilitating more secure and private transactions.

The essence of threshold signatures is explored, stressing the balance required between securing funds and avoiding self-lockout. This balance is crucial in determining the most effective threshold level that minimizes expected loss from both potential attacks and the risk of losing access to one’s funds. A formal model presented introduces a way to calculate the optimal threshold, $\tau^{*}$, taking into account the differential impacts of security parameters on user and attacker probability functions. The analysis extends to dynamic scenarios, accounting for evolving conditions over time that affect access probabilities and, consequently, optimal threshold adjustments and timelock settings.

Significant outcomes of this research include a conceptual framework for understanding and optimizing threshold signatures in the context of Bitcoin, especially following the Taproot upgrade. This framework not only contributes to the academic discourse on cryptocurrency security but also opens avenues for practical applications, including the integration of AI agents in complex transaction structures. For further insights and practical applications, the source code for simulations discussed is made available at a GitHub repository. This work illuminates the critical considerations in setting optimal thresholds for Bitcoin transactions and highlights Taproot's potential in supporting diverse economic activities through advanced contract structures.